Another one is that there's an infinite set of infinites between any two numbers. As in, between 1 and 2, there are an infinite amount of decimal numbers. So, we can think of the theoretical distance between two sets of whole numbers as two infinite universes. But that seems silly, right?

And let's say we compare the infinity between 1 and 2 to the infinity between 1 and 3. The infinity between 1 and 3 contains the infinity between 1 and 2

The accountant one is dumb, and is more a trick question than any sort of mind-boggling concept in math

And these are nowhere near the most controversial concepts in math, aside from maybe those which deal with infinity.

the monty hall problem is quite simple, really. When presented in this way, it just sounds more confusing with 3 choices.

But let me put it in a somewhat different light that makes more sense...

Suppose there is a lottery; you need to pick a # from 000 to 999. There are 1,000 possible number combinations here.

If you guess the correct number, let's say you win 900x your bet, leaving a modest house edge of 10% (better than a state lottery, which is usually like 35-50%)

I guess one set of numbers, let's say 351. So i have a 1 in 1,000 chance of winning, or so it seems.

However, someone from the lottery (who has all sorts of inside information and knows the winning number) says to me "Brendan, i will eliminate 600 possible combinations of numbers and let you change it to 932, or you can keep your original pick of 351"

so i can either change this to 932, or keep it at 351... it would only make sense for me to change to 932, because he just told me he eliminated 600 possible numbers- and 932 is not one of them.

Suddenly, my odds of winning go from 1 in 1,000 to 1 in 400

This relates to the 3 door problem, BECAUSE THE DOOR HE OPENS DOES NOT HAVE THE PRIZE- so you know it cant be the right one. Similarly, the 600 numbers eliminated, do not hold the winning combination!

Rhi_Bran saidAnother one is that there's an infinite set of infinites between any two numbers. As in, between 1 and 2, there are an infinite amount of decimal numbers. So, we can think of the theoretical distance between two sets of whole numbers as two infinite universes. But that seems silly, right?

And let's say we compare the infinity between 1 and 2 to the infinity between 1 and 3. The infinity between 1 and 3 contains the infinity between 1 and 2

The accountant one is dumb, and is more a trick question than any sort of mind-boggling concept in math

And these are nowhere near the most controversial concepts in math, aside from maybe those which deal with infinity.

The paradox like situation arises due to not understanding the concept.Yep there are infinite numbers between one and two, physically that means if two points are one unit,(lets say meter) apart then there are infinite points between them, whats paradoxical about that? If you go on magnifying it no matter how close two point are you can still have another point between them. For eg if two pinheads are in contact, if you magnify to atomic level there's still the point where the two atoms of both the pins touch. Even if you take two ponts inside the atom you can still fit in a lots of nuclei. If the two points are inside the nucleus then also the barons can be inserted. So no matter how close two points are magnify them and you can have more points between them. This doesnt mean that the 1meter distance we started with has become too big or is infinte.

Rhi_Bran saidAnd these are nowhere near the most controversial concepts in math, aside from maybe those which deal with infinity.

To be clear: these aren't controversial concepts in math at all. They are mathematical concepts (except Benford's law, which is a statistical observation) that are controversial among the larger populace. They're all quite simple mathematically, merely non-intuitive for many.

Also, each of the problems was nicely explained except Benford's law. In part this is because it isn't a "mathematical fact". It's a statistical observation. It does however have some presumptive explanations. I couldn't find anything equally simple to the rest of what was presented here, but in case anyone wanted:http://www.statisticalconsultants.co.nz/blog/B43.html

brendanmuscles saidSuddenly, my odds of winning go from 1 in 1,000 to 1 in 400

This relates to the 3 door problem, BECAUSE THE DOOR HE OPENS DOES NOT HAVE THE PRIZE- so you know it cant be the right one. Similarly, the 600 numbers eliminated, do not hold the winning combination!

Exactly. This is the part that used to spin me around. With two doors remaining, it seems like the odds in the here and now are 50-50 so why would you switch? It's because the omnipotent game show host is able to elimate a door known to be wrong. You picked your door out of 3 that could have been right or wrong.

Rhi_Bran saidAnd these are nowhere near the most controversial concepts in math, aside from maybe those which deal with infinity.

To be clear: these aren't controversial concepts in math at all. They are mathematical concepts (except Benford's law, which is a statistical observation) that are controversial among the larger populace. They're all quite simple mathematically, merely non-intuitive for many.

Also, each of the problems was nicely explained except Benford's law. In part this is because it isn't a "mathematical fact". It's a statistical observation. It does however have some presumptive explanations. I couldn't find anything equally simple to the rest of what was presented here, but in case anyone wanted:http://www.statisticalconsultants.co.nz/blog/B43.html

[quote] ... The paradox like situation arises due to not understanding the concept. ... [/quote]

Epistemology is a dialectic.

'One' in the English language is a metaphor for a discrete unit and a union of all discrete units, simultaneously. Its corollary in material form is perhaps the un-orientable objects like the Mobius Strip or Klein Bottle.